Oh I hear you, wet season and all...
Sounds just like my little girl, although she was at that stage at around 8 months. These are a few ideas of things that have helped me over the past couple of months to deal with a very inquisitive explorer in a small apartment with limited opportunities for outside play:
- Make an obstacle course with chairs, small tables etc that he can 'cruise' along from one end to the other, perhaps with small toys spaced along the way
Do you have any floor length curtains? Playing with and hiding behind curtains is good amusement here.
Ball activities: catching, crawling after, and kicking. My LO is an ace footballer already, if I'm holding her hands
Also rolling a ball through a tunnel or down a ramp is lots of fun.
Water activities: put a big bowl of water on a towel on the kitchen or bathroom floor, and let him splash around in it.
At around nine months she got really interested in finding things 'hidden' under a box, in a bag, etc. I line up a row of containers with a small object hidden under each one and she crawls along and tips each one over.
Can you find a local cafe or similar that is another safe indoor space for him to explore? I found a place nearby that has a floor to ceiling mirror wall, which F LOVES. I go there when I know it'll be nice and quiet so she's not under anyone's feet, and she can cruise (or now toddle) around the chairs and under the tables next to the mirror wall. And I can sit and have a coffee in peace.
Teach him baby sign language a) so in a few months he can communicate what he wants to play with but also b) it becomes a fun game in itself!
Action songs and rhymes.
Put on some music and have a good dance together.
Good idea about the stairs, especially teaching him how to get down again. We try to preempt things like that too - so I know it's only a matter of time until she figures out how to climb onto our bed, so I'm already teaching her her to get down from it!
F also loves going for walks in the rain, in the frontpack - don't know what your rainy season is like, but ours was mostly fairly vertical rain, not a lot of wind, so with a good umbrella she'd stay dry. If it was a bit windy, I'd put DH's jacket on over both of us

. She loves watching the rain drip off things, hearing the sound on the umbrella, etc. And good to get a bit of fresh air everyday.
HTH xx